Fa'ata'ita'iga o Fesili e Talanoaina ai le Fa'aopoopoga o Vector e ala i le Component
O le fa'aopoopoga o vector o se faiga fa'avae i le fisiki ma le matematika e fa'aaogaina e su'e ai le taunuuga o ni vector se lua pe sili atu. O le auala fa'apitoa i le fo'ia o le fa'aopoopoga o vector o se metotia aoga tele, aemaise lava pe a feagai ma vectors i itu e lua pe tolu. O lenei tusiga o le a fa'amatalaina ai le manatu o le fa'aopoopoga o vector i itu e lua ma tu'uina atu ni fa'ata'ita'iga o fa'afitauli ma fofo.
O le Manatu o le Fa'aopoopoga o Vector Vaega
O vectors uma i le avanoa lua-dimensional (2D) e mafai ona vaevaeina i ni vaega se lua: o le vaega x (fa'alava) ma le vaega y (tu sa'o). I le tolu-dimensional (3D), o vectors e iai se isi vaega, o le vaega z (loloto).
Faapea e lua a tatou vectors A ma le B. O vaega o nei vectors e mafai ona faʻaalia e pei ona taua i lalo:
– O le Vector A e iai vaega \(A_x\) ma le \(A_y\) i le 2D (pe fa'apea fo'i ma le \(A_z\) i le 3D).
– O le Vector B e iai vaega \(B_x\) ma le \(B_y\) i le 2D (pe fa'apea fo'i ma le \(B_z\) i le 3D).
O le fa'aopoopoina o nei vectors e lua o le a maua ai se vector R e maua mai ai vaega nei:
\[ R_x = A_x + B_x \]
\[ R_y = A_y + B_y \]
Mo vectors i le 3D, o le vaega z e fa'apea fo'i:
\[ R_z = A_z + B_z \]
A maeʻa ona faʻatatauina vaega taʻitasi o le vector taunuuga, e mafai ona tatou maua le modulus (tele) ma le itu o le vector taunuuga e faʻaaoga ai le fua faʻatatau:
\[ |R| = \sqrt{R_x^2 + R_y^2} \] (mo le 2D)
Pe mo le 3D:
\[ |R| = \sqrt{R_x^2 + R_y^2 + R_z^2} \]
Ma o le itu e agai i ai le vector e mafua mai ai e mafai ona fuafuaina e ala i le tulimanu i le coordinate axes.
Fesili Fa'ata'ita'i ma Talanoaga
Fesili 1
I le tuuina atu o ni vectors se lua i se vaalele lua-dimensional:
– O le A e \(5 \, \text{unit}\) i sasa'e.
– O B o lo'o i le \(3 \, \text{unit}\) i matū.
Fuafua le vector R e maua mai ai.
Talanoaga
Muamua, tatou te liua le vector i ona vaega taʻitasi.
– Vector A : \(A = (5, 0)\) auā e na'o le x lona vaega.
– Vector B : \(B = (0, 3)\) auā e na'o le vaega y e iai.
O le aofaʻi lea o vaega:
\[ R_x = A_x + B_x = 5 + 0 = 5 \]
\[ R_y = A_y + B_y = 0 + 3 = 3 \]
Ona o le vector R lea e mafua mai i le taunuuga:
\[ R = (5, 3) \]
Mo le fuafuaina o le umi (modulus) o le vector R:
\[ |R| = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 \]
E mafai ona fuafuaina le itu o le vector R e faʻaaoga ai le tulimanu θ i le x-axis:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{3}{5} \]
\[ \theta = \arctan\left(\frac{3}{5}\right) \approx 30.96^\circ \]
O lea la, o le vector R e mafua mai i ai lona umi e tusa ma le 5.83 iunite ma e fausia ai se tulimanu e 30.96° ma le x-axis.
Fesili 2
Tuuina atu ni vectors se lua i fua e tolu:
– O le A o le \(3\hat{i} + 2\hat{j} + 1\hat{k}\)
– O le B o le \(1\hat{i} + 4\hat{j} + 2\hat{k}\)
Fuafua le vector R e maua mai ai.
Talanoaga
Muamua, matou te faʻailoaina vaega o vectors taʻitasi:
– Veketo A : \(A_x = 3\), \(A_y = 2\), \(A_z = 1\).
– Veketa B : \(B_x = 1\), \(B_y = 4\), \(B_z = 2\).
O le aofaʻi lea o vaega:
\[ R_x = A_x + B_x = 3 + 1 = 4 \]
\[ R_y = A_y + B_y = 2 + 4 = 6 \]
\[ R_z = A_z + B_z = 1 + 2 = 3 \]
Ona o le vector R lea e mafua mai i le taunuuga:
\[ R = (4, 6, 3) \]
Mo le fuafuaina o le umi (modulus) o le vector R:
\[ |R| = \sqrt{4^2 + 6^2 + 3^2} = \sqrt{16 + 36 + 9} = \sqrt{61} \approx 7.81 \]
E mafai ona fuafuaina le itu o le vector R e faʻatatau i le x, y, ma le z axes e faʻaaoga ai le cosine o le director:
\[ \cos(\alpha) = \frac{R_x}{|R|} = \frac{4}{7.81} \approx 0.512 \]
\[ \alpha = \arccos(0.512) \approx 59.50^\circ \]
\[ \cos(\beta) = \frac{R_y}{|R|} = \frac{6}{7.81} \approx 0.768 \]
\[ \beta = \arccos(0.768) \approx 39.50^\circ \]
\[ \cos(\gamma) = \frac{R_z}{|R|} = \frac{3}{7.81} \approx 0.384 \]
\[ \gamma = \arccos(0.384) \approx 67.64^\circ \]
O lea la, o le vector R e mafua mai i ai lona umi e tusa ma le 7.81 iunite ma o ona itu e faatatau i le x, y, ma le z axes e 59.50°, 39.50°, ma le 67.64°.
Fesili 3
Tuuina atu ni vectors se lua:
– O le telē o le P e 4 iunite ma e fausia ai se tulimanu e 45° i le x-axis lelei.
– O le telē o le Q e 6 iunite ma e fausia ai se tulimanu e 120° i le x-axis lelei.
Fuafua le vector R e maua mai ai.
Talanoaga
Muamua, tatou vaevaeina le vector i ona vaega x ma le y:
– Veketa P : \(P_x = 4\cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\), \(P_y = 4\sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\).
– Veketa Q : \(Q_x = 6\cos(120^\circ) = 6 \cdot \left(-\frac{1}{2}\right) = -3\), \(Q_y = 6\sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} \approx 5.2\).
O le aofaʻi lea o vaega:
\[ R_x = P_x + Q_x = 2.83 – 3 = -0.17 \]
\[ R_y = P_y + Q_y = 2.83 + 5.2 = 8.03 \]
Ona, o le vector R e mafua mai i le:
\[ R = (-0.17, 8.03) \]
Mo le fuafuaina o le umi (modulus) o le vector R:
\[ |R| = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.0289 + 64.48} = \sqrt{64.509} \approx 8.03 \]
Itu o le vector R:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{8.03}{-0.17} = -47.24 \]
\[ \theta = \arctan(-47.24) \approx -88.99^\circ \]
Peita'i, o lenei tulimanu e fuaina e uiga i le x-axis leaga, o lea la o le tulimanu moni i le tulaga o le fa'afitauli o le:
\[ 180^\circ – 88.99^\circ \pe tusa ma le 91.01^\circ \]
O lea la, o le vector R e mafua mai i ai lona umi e tusa ma le 8.03 iunite ma e fausia ai se tulimanu e 91.01° faatasi ai ma le x-axis lelei.
Ua talanoaina i lenei tusiga le fa'aopoopoga o vector i vaega, ma tu'uina atu ai ni fa'ata'ita'iga o fa'afitauli ma fofo. E matua aoga tele le metotia i vaega i le fa'afaigofieina o fa'atusatusaga ma le tu'uina atu o se auala fa'atulagaina e foia ai fa'afitauli vector i le fua fa'amatematika o le avanoa.
